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Ordinary Differential Equations and Dynamical Systems

  • Book
  • © 2013

Overview

Authors:
  1. Thomas C. Sideris

    1. Department of Mathematics, University of California, Santa Barbara, USA

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  • Proofs are presented with complete mathematical details
  • Extensive use of the implicit function theorem as a unifying approach to perturbative analysis
  • Exercises are included at the end of each chapter permit use as a textbook

Part of the book series: Atlantis Studies in Differential Equations (ASDE, volume 2)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Introduction

    • Thomas C. Sideris
    Pages 1-3

  3. Linear Systems

    • Thomas C. Sideris
    Pages 5-19

  4. Existence Theory

    • Thomas C. Sideris
    Pages 21-51

  5. Nonautonomous Linear Systems

    • Thomas C. Sideris
    Pages 53-72

  6. Results from Functional Analysis

    • Thomas C. Sideris
    Pages 73-87

  7. Dependence on Initial Conditions and Parameters

    • Thomas C. Sideris
    Pages 89-94

  8. Linearization and Invariant Manifolds

    • Thomas C. Sideris
    Pages 95-118

  9. Periodic Solutions

    • Thomas C. Sideris
    Pages 119-154

  10. Center Manifolds and Bifurcation Theory

    • Thomas C. Sideris
    Pages 155-198

  11. The Birkhoff Smale Homoclinic Theorem

    • Thomas C. Sideris
    Pages 199-217

  12. Back Matter

    Pages 219-225
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Keywords

About this book

This book is a mathematically rigorous introduction to the beautiful subject of ordinary differential equations for beginning graduate or advanced undergraduate students. Students should have a solid background in analysis and linear algebra. The presentation emphasizes commonly used techniques without necessarily striving for completeness or for the treatment of a large number of topics. The first half of the book is devoted to the development of the basic theory: linear systems, existence and uniqueness of solutions to the initial value problem, flows, stability, and smooth dependence of solutions upon initial conditions and parameters. Much of this theory also serves as the paradigm for evolutionary partial differential equations. The second half of the book is devoted to geometric theory: topological conjugacy, invariant manifolds, existence and stability of periodic solutions, bifurcations, normal forms, and the existence of transverse homoclinic points and their link to chaotic dynamics. A common thread throughout the second part is the use of the implicit function theorem in Banach space. Chapter 5, devoted to this topic, the serves as the bridge between the two halves of the book.

Reviews

From the book reviews:

“This is a very nice text for a beginners course on ordinary differential equations and dynamical systems. … All basic results about ordinary differential equations are present. … Each chapter contains examples and exercises … . this book will certainly be an excellent reference text.” (Santiago Ibáñez, Mathematical Reviews, December, 2014)

Authors and Affiliations

  • Department of Mathematics, University of California, Santa Barbara, USA

    Thomas C. Sideris

Bibliographic Information

  • Book Title: Ordinary Differential Equations and Dynamical Systems

  • Authors: Thomas C. Sideris

  • Series Title: Atlantis Studies in Differential Equations

  • DOI: https://doi.org/10.2991/978-94-6239-021-8

  • Publisher: Atlantis Press Paris

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature B.V. 2013

  • Hardcover ISBN: 978-94-6239-020-1Published: 30 October 2013

  • eBook ISBN: 978-94-6239-021-8Published: 17 October 2013

  • Series ISSN: 2214-6253

  • Series E-ISSN: 2214-6261

  • Edition Number: 1

  • Number of Pages: XI, 225

  • Number of Illustrations: 11 b/w illustrations

  • Topics: Ordinary Differential Equations

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